Question

From the Wiener process extract the subensemble corresponding to \(Y\left(t_{0}\right)=y_{0}\). Find the evolution of \(\langle Y(t)\rangle\) for \(t>t_{0}\). Also find the variance \(\left\langle\left\langle Y(t)^{2}\right\rangle\right)\). in this ensemble.

Short answer

The expected value of \(Y(t)\) for \(t > t_{0}\) is \(y_{0}\), and the variance of \(Y(t)\) for \(t > t_{0}\) is \(t - t_{0}\).

Step-by-step solution

- Understanding the Wiener process

The Wiener process, denoted as W(t), is a mathematical model of Brownian motion. It has the following properties: W(0) = 0, it has independent and normally distributed increments, and a mean of 0 and a variance of t for any increment W(t) - W(s) where t > s. In this exercise, we examine a subensemble of the Wiener process where the process takes a specific value, y_0, at time t_0. This subensemble can be represented as Y(t) such that Y(t_0) = y_0.

- Calculating the expected value of Y(t)

To find the expected value of Y(t), denoted as \(\langle Y(t) \rangle\), for t > t_0, we need to consider that for any time t > t_0, Y(t) is normally distributed with mean y_0 and variance \(t - t_0\), because the increments after \(t_0\) are independent of the past and have a distribution centered at zero. Therefore, the expected value of Y(t) remains y_0 for all t > t_0.

- Calculating the variance of Y(t)

The variance of the Wiener process at any time t is given by the time t itself. Since the independent increments for the Wiener process start at time t_0 with a value of y_0, the variance of Y(t) for t > t_0 is simply the duration \(t - t_0\), which is the variance of the increment from time t_0 to t.

Key concepts

Brownian motion

Brownian motion, a phenomenon first observed by botanist Robert Brown in 1827, refers to the erratic and continuous movement of small particles suspended in a fluid. This motion results from collisions with the surrounding fluid molecules. In the context of finance and mathematics, Brownian motion represents a model for randomness and chaos found in various systems.

Brownian motion is widely used to model the unpredictable and stochastic nature of particle paths and has extensive applications in physics, biology, and finance, especially in the pricing of financial derivatives. In our textbook problem, Brownian motion is characterized mathematically as a Wiener process, serving as a fundamental building block in stochastic processes.

Stochastic processes

Stochastic processes encompass a collection of random variables representing the evolution of some system over time in a probabilistic manner. They are used to model seemingly random or unpredictable phenomena in various fields, from physics to finance to social sciences.

The Wiener process, an example of a stochastic process, is central to the study of Brownian motion and serves as a benchmark for 'no memory' and 'continuous-time' stochastic processes. As shown in our exercise, specific characteristics of a subensemble of the Wiener process, a subset of the larger stochastic process, can be scrutinized to understand the system's dynamics following an initial condition, such as the particle being at a particular position at a given time.

Expected value

Expected value is a fundamental concept in probability and statistics, representing the average outcome of a random variable if an experiment is repeated many times. More formally, it is the weighted average of all possible values, with the weights being the probabilities of occurrence.

In the context of the Wiener process, if we focus on a subensemble starting at a known value at a given time, the expected value remains constant for all future times, as shown in our step-by-step solution. Even as the process advances, since new increments are normally distributed and centered at zero, the expected value doesn't deviate from the initial value at the time we started observing the process.

Variance

Variance is a measure of the spread of a set of numbers, reflecting how much the numbers differ from the expected value or mean. In statistical terms, it quantifies the dispersion of a random variable's possible outcomes and is calculated as the average of the squared differences from the mean.

For our Wiener process exercise, the variance represents the degree of uncertainty or risk about the position of the particle as time progresses. As we established in the solution, the variance for the subensemble of this stochastic process increases linearly with time, beginning from our initial observation point, showing that uncertainty grows as we project further into the future from time t_0.